It was conjectured by Gaiotto, that (quantized) K-theoretic Coulomb branches of 3d \mathcal{N} =4 SUSY gauge theories, as defined by Braverman, Finkelberg, and Nakajima, bear structure of (quantum) cluster varieties. I will outline a proof of this conjecture for quiver gauge theories, provided that the quiver does not have loops. In relation to this conjecture, I will also discuss why positive and Gelfand-Tsetlin representations of quantum groups are unitary equivalent, and how to construct higher rank Fenchel–Nielsen coordinates on moduli spaces of PGL_n local systems. This is a joint work with Gus Schrader.